A080042 -id:A080042 - cp's OEIS frontend

A015530 Expansion of x/(1 - 4x - 3x^2).

0, 1, 4, 19, 88, 409, 1900, 8827, 41008, 190513, 885076, 4111843, 19102600, 88745929, 412291516, 1915403851, 8898489952, 41340171361, 192056155300, 892245135283, 4145149007032, 19257331433977, 89464772757004

Offset: 0

Olivier Gérard

Let b(1)=1, b(k) = floor(b(k-1)) + 3/b(k-1); then for n>1, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002
In general, x/(1 - a*x - b*x^2) has a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1,k)*b^k*a^(n-2k-1). - Paul Barry, Apr 23 2005
Pisano period lengths: 1, 2, 1, 4, 24, 2, 21, 4, 3, 24, 40, 4, 84, 42, 24, 8, 288, 6, 18, 24, ... . - R. J. Mathar, Aug 10 2012
This is the Lucas sequence U(4,-3). - Bruno Berselli, Jan 09 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (4,3).

Appears in A179596, A126473 and A179597. - Johannes W. Meijer, Aug 01 2010

Cf. A080042: Lucas sequence V(4,-3).

I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2012

LinearRecurrence[{4,3},{0,1},30] (* Vincenzo Librandi, Jun 19 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-4*x-3*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,4,-3) for n in range(0, 23)]# Zerinvary Lajos, Apr 23 2009

a(n) = 4*a(n-1) + 3*a(n-2).
a(n) = (A086901(n+2) - A086901(n+1))/6. - Ralf Stephan, Feb 01 2004
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*3^k*4^(n-2k-1). - Paul Barry, Apr 23 2005
a(n) = ((2+sqrt(7))^n - (2-sqrt(7))^n)/sqrt(28). Offset 1. a(3)=19. - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = A108851(n)+a(n)*sqrt(7).
Limit(A108851(n)/a(n), n=infinity) = sqrt(7). (End)
G.f.: x*G(0) where G(k)= 1 + (4*x+3*x^2)/(1 - (4*x+3*x^2)/(4*x + 3*x^2 + 1/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 28 2012
G.f.: G(0)*x/(2-4*x), where G(k)= 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013

A108851 a(n) = 4a(n-1) + 3a(n-2), a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, 10878914, 50540729, 234799658, 1090820819, 5067682250, 23543191457, 109375812578, 508132824683, 2360658736466, 10967033419913, 50950109889050

Offset: 0

Philippe Deléham, Jul 11 2005

Binomial transform of A083098, second binomial transform of (1, 0, 7, 0, 49, 0, 243, 0, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,3).

Cf. A080042. - Zerinvary Lajos, May 14 2009
Cf. A015530, A083098.
Appears in A179596, A179597 and A126473. - Johannes W. Meijer, Aug 01 2010

[Floor(((2 + Sqrt(7))^n + (2 - Sqrt(7))^n) / 2): n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

LinearRecurrence[{4,3},{1,2},30] (* Harvey P. Dale, Jan 02 2022 *)

a(n)=round(((2+sqrt(7))^n+(2-sqrt(7))^n)/2) \\ Charles R Greathouse IV, Dec 06 2011

[lucas_number2(n,4,-3)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

a(n) = ((2 + sqrt(7))^n + (2 - sqrt(7))^n) / 2.
G.f.: (1 - 2*x) / (1 - 4*x - 3*x^2).
E.g.f.: exp(2*x)*cosh(sqrt(7)*x).
a(n+1)/a(n) converges to 2 + sqrt(7) = 4.645751311064...
Limit_{k->oo} a(n+k)/a(k) = A108851(n) + A015530(n)*sqrt(7); also lim_{n->oo} A108851(n)/A015530(n) = sqrt(7). - Johannes W. Meijer, Aug 01 2010
a(n) = Sum_{k=0..n} A201730(n,k)*6^k. - Philippe Deléham, Dec 06 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = (2 + sqrt(7))^n - A015530(n)*sqrt(7). - Robert FERREOL, Aug 04 2025

A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,    0,     0,      0,       0,        0,        0, ...
1, 1,  2,   4,    8,   16,    32,     64,     128,      256,      512, ...
1, 1,  3,   7,   17,   41,    99,    239,     577,     1393,     3363, ...
1, 1,  4,  10,   28,   76,   208,    568,    1552,     4240,    11584, ...
1, 2,  8,  32,  128,  512,  2048,   8192,   32768,   131072,   524288, ...
1, 2,  9,  38,  161,  682,  2889,  12238,   51841,   219602,   930249, ...
1, 2, 10,  44,  196,  872,  3880,  17264,   76816,   341792,  1520800, ...
1, 2, 11,  50,  233, 1082,  5027,  23354,  108497,   504050,  2341691, ...
1, 2, 12,  56,  272, 1312,  6336,  30592,  147712,   713216,  3443712, ...
1, 3, 18, 108,  648, 3888, 23328, 139968,  839808,  5038848, 30233088, ...
1, 3, 19, 117,  721, 4443, 27379, 168717, 1039681,  6406803, 39480499, ...
1, 3, 20, 126,  796, 5028, 31760, 200616, 1267216,  8004528, 50561600, ...
1, 3, 21, 135,  873, 5643, 36477, 235791, 1524177,  9852435, 63687141, ...
1, 3, 22, 144,  952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().

T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019

T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019

T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

A080043 a(n)=floor((2+sqrt(7))^n).

Original entry on oeis.org

1, 4, 21, 100, 465, 2164, 10053, 46708, 216993, 1008100, 4683381, 21757828, 101081457, 469599316, 2181641637, 10135364500, 47086382913, 218751625156, 1016265649365, 4721317472932, 21934066839825, 101900219778100

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 21 2003

Index entries for linear recurrences with constant coefficients, signature (4, 4, -4, -3).

CoefficientList[Series[(1+t^2+4t^3)/(1-4t-4t^2+4t^3+3t^4), {t, 0, 25}], t]
With[{c=2+Sqrt[7]},Floor[c^Range[0,30]]] (* or *) LinearRecurrence[{4,4,-4,-3},{1,4,21,100},30]

G.f.: g(t)=(1+t^2+4t^3)/(1-4t-4t^2+4t^3+3t^4) a(n)=A080042(n)-(1+(-1)^n)/2

a(0)=1, a(1)=4, a(2)=21, a(3)=100, a(n)=4*a(n-1)+4*a(n-2)- 4*a(n-3)- 3*a(n-4). - Harvey P. Dale, Aug 11 2015

cp's OEIS Frontend

A015530 Expansion of x/(1 - 4*x - 3*x^2).

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A108851 a(n) = 4*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 2.

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A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

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A080043 a(n)=floor((2+sqrt(7))^n).

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A015530 Expansion of x/(1 - 4x - 3x^2).

A108851 a(n) = 4a(n-1) + 3a(n-2), a(0) = 1, a(1) = 2.